Thursday, August 27, 2009

Extra LIGO

1 Comments:

Figure 1 in the LOGO paper you link shows that they're looking for gravitational waves at frequencies from 30 Hz (where the frequency axis begins) to 4 kHz.

The implosion process which involves the most massive accelerations (and hence the greatest radiating power for gravitational waves) in supernova explosions takes t = 0.2 second, so the main frequency is going to be on the order f = 1/t = 5 Hz.

Also, I don't see the point in making using refined formulae to predict the precise shape of the gravitational waves, when so far they can't detect anything at all! All you need is the estimate the amount of energy and the frequency, just as when forecasting an earthquake. Don't worry about trying to predict the exact shape of the waveform, just worry whether your instrument is going to pick up enough energy at the right frequency to give a reading or not.

From GR: P = [G*(ma)^2]/(2*Pi*c^3)

watts of gravitational wave power from mass m having acceleration a. [Most textbooks give a formula for two orbiting masses radiating gravitational waves, but they are accelerating in their orbit with acceleration a = (v^2)/r, so can be simplified easily to the formula above.]

For a Type II supernova, the critical mass is 1.38 solar masses hence m = 2.7*10^30 kg. The initial radius of the largest white dwarf (before collapse and supernova) is 0.02 solar radii = 1.4*10^7 m. The white dwarf collapses down to 30 km radius at speeds reaching 0.23c or 7*10^7 m/s which takes 0.2 second, when it rebounds and the ~10^44 J supernova explosion phase then occurs. The mean acceleration over the 0.2 seconds of implosion is a = v/t = (7*10^7)/0.2 = 3.5*10^8 ms^-2. Inserting this acceleration and mass into the equation, the mean gravitational wave radiating power is 3.7*10^41 W, multiplying by the 0.2 second duration gives E = 7.4*10^40 J for the energy in gravitational waves.

This is merely 0.074% of the ~10^44 J of explosive energy in the Type II supernova! It's a tiny amount of energy and the detector must be tined to the right frequency to have any chance. The mean frequency is obviously on the order of 1/t = 5 Hz.

If we assume isotropic emission of this 7.4*10^40 J GW energy over area 4*Pi*R^2, then a even a relatively nearby supernova on the other side of the galaxy (100,000 light-years or 9.5*10^20 m away) will only give us 0.0065 J/m^2 of grvitational wave energy. Acting over a pulse period of 0.2 second.

E = F*x = (ma)*(0.5at^2) = (1/2)*m*(at)^2

So a 1000 kg cube of water with sides 1 m long will receive 0.0065 J and will pick up an acceleration of 0.018 ms^-2. You would think that they could detect this kind of acceleration, if they searched at the right frequency!